Optimal Risk Sharing with Different Reference Probabilities
Beatrice Acciaio∗
Gregor Svindland†
January 12, 2009
Abstract
We investigate the problem of optimal risk sharing between agents endowed with cashinvariant choice functions which are law-invariant with respect to different reference probability measures. We motivate a discrete setting both from an operational and a theoretical
point of view, and give sufficient conditions for the existence of Pareto optimal allocations
in this framework. Our results are illustrated by several examples.
Keywords: Optimal risk sharing, law-invariance, convolution
IME Subject and Insurance Branch Category: IM51, IE12
JEL classification: D81, G22
MSC 2000: 91B06, 91B16, 91B30
1
Introduction
The optimal exchange of risk between two parties is one of the major issues in mathematical
economics and finance, and many authors have studied this problem, since the early works of
Arrow [2], Borch [5], Du Mouchel [11], where the risk sharing is analyzed in the insurance and
reinsurance context. The introduction of concepts like coherent and convex risk measures, by
Artzner et al. [3] and Föllmer and Schied [17], recently paved the way to a further analysis
of the problem of optimal sharing and allocation of risk. Several authors have considered
the exchange of risk between agents endowed with these kinds of choice functions (see, e.g.,
Deprez and Gerber [9], Chateauneuf et al. [8], Barrieu and El Karoui [4], Jouini et al. [24],
∗
Financial and Actuarial Mathematics, Vienna University of Technology, Wiedner Hauptstrasse 8-10/105-1,
A-1040 Vienna, Austria, [email protected]
†
Mathematics Institute, University of Munich, Theresienstrasse 39, D-80333 München, Germany,
[email protected]
1
Filipović and Kupper [13, 14], Burgert and Rüschendorf [6, 7]) or in a slightly more general
setting (see, e.g., Acciaio [1], Filipović and Svindland [16]).
A natural property to require on those choice functions is indifference with respect to
financial positions having the same distribution under some reference probability measure.
This is the so-called law-invariance property, studied, e.g., by Kusuoka [25], Frittelli and
Rosazza Gianin [19], Jouini et al. [23]. When all choice functions are assumed to be cashinvariant (or translation invariant) and law-invariant with respect to the same reference
probability measure, the existence of optimal allocations has already been proved, see Jouini
et al. [24], Filipović and Svindland [16] and Acciaio [1]. In this paper we study the risk
sharing problem in the situation of two economic agents with different views of the world,
that is, with different (subjective) reference probability measures. We consider them equipped
with cash-invariant choice functions which are law-invariant in their respective worlds, i.e.,
with respect to their different reference probabilities. Manifold causes may motivate such a
framework. In case of financial corporations, for instance, these different world views might
stem from different internal models, from having access to different information, or from
being subject to guidelines of different regulating agencies. We can consider, for example,
the case of financial firms subject to stress tests, possibly managed by different supervising
agencies. These tests are made to gauge the potential vulnerability of the firms to a given
set of particular market events or stress scenarios, like a stock market crash or other market
shocks. In this situation firms are interested in maintaining a ‘good’ position in case one of
those shock scenarios were to occur, in order to ‘pass’ the test. This might amongst other
things be achieved by exchanging risk in this set of events. We like to point out that our
setting is in particular valid in a situation of both competition and cooperation. It is often
observed that competing agents in a market do cooperate on the level of exchanging certain
risks if they all believe to benefit from it. However, since in the end the agents are competitors,
it is very unlikely that they will share full information with each other, that is agree on one
reference probability, and thereby abandon advantages over their competitors which might
result from access to some particular information. E.g. in insurance companies or banks the
internal model is a crucial competitive advantage which is kept secret from other agents in
the market. Within one firm, i.e. within one internal model, law-invariance of the choice
function used is very likely. Hence, we end up in a situation as described above where the
incomplete and different information is represented by the different reference probabilities of
the agents which are a priori unrelated. Nevertheless, these competing agents may all benefit
from trading/exchanging some of their risks on some prominent events which are common
knowledge, like increase or decrease of interests, natural hazards, etc., but which are weighted
differently according to different information.
In any case, any exchange of risk requires some kind of cooperation between the involved
agents, who will have to agree on some set of (prominent) scenarios on which they want some
2
level of protection. As described above it is very likely that the agents weight these scenarios
in a different way depending on their respective information. Moreover, it turns out that, in
practice, such a set of scenarios is usually finite, thus reducing the optimal risk allocation
problem to a finite dimensional risk exchange on these base scenarios. In this framework, and
under some mild additional conditions, we show that there always exist Pareto optimal allocations for any aggregate risk. Our results are illustrated by several examples. In particular,
we also give examples which show that Pareto optimal allocations do not exist in general if
we drop some of our assumptions (Examples 4.3 and 4.4).
The remainder of the paper is organized as follows. In Section 2 we formalize the optimal
risk sharing problem, and we state our main result on the existence of optimal allocations
(Theorem 2.3). This result is then proved in Section 3. Our examples are collected in Section 4. We assume the reader to be familiar with basic convex duality as outlined in [26] or
[12]. However, in the appendix we give a short review on some basic concepts and notation
from convex analysis which are frequently used throughout the paper. Some known results
are also postponed to the appendix.
2
2.1
Optimal Risk Sharing Problem
Framework
We consider a measurable space (Ω, F) and two probability measures P1 , P2 on (Ω, F) such
that (Ω, F, Pi ), i = 1, 2, are non-atomic standard probability spaces. The measures P1 , P2
describe the view of two agents, say 1 and 2, on the world (Ω, F). The preferences of the
i-th agent on L∞ (Ω, F, Pi ) are represented by a choice function Ui : L∞ (Ω, F, Pi ) → R, that
throughout the paper is assumed to satisfy the following conditions:
(C1) concavity: Ui (αX + (1 − α)Y ) ≥ αUi (X) + (1 − α)Ui (Y ) for all X, Y ∈ L∞ (Ω, F, Pi )
and α ∈ (0, 1);
(C2) cash-invariance: Ui (X + c) = Ui (X) + c for all X ∈ L∞ (Ω, F, Pi ) and c ∈ R;
(C3) normalization: Ui (0) = 0;
(C4) Pi -law-invariance: Ui (X) = Ui (Y ) whenever X, Y ∈ L∞ (Ω, F, Pi ) are identically distributed under Pi ;
(C5) upper semi-continuity (u.s.c.): for any sequence (Xn )n∈N ⊂ L∞ (Ω, F, Pi ) converging to
some X ∈ L∞ , we have Ui (X) ≥ lim supn Ui (Xn ).
If Ui in addition is monotone, i.e. Ui (X) ≥ Ui (Y ) whenever X, Y ∈ L∞ (Ω, F, Pi ) satisfy X ≥
Y Pi -a.s., then Ui is a Pi -law-invariant monetary utility function, i.e. −Ui is a Pi -law-invariant
convex risk measure in the sense of Föllmer and Schied [18]. Note that (C5) is equivalent
to the continuity of Ui because Ui is finitely-valued (see e.g. [12] Corollary 2.5), and that
3
any proper function on L∞ (Ω, F, Pi ) which is monotone and satisfies (C2) is automatically
finitely-valued and 1-Lipschitz-continuous (see [18]). It is proved in [23] that the Pi -lawinvariance ensures the following dual representation for Ui (the so-called Fatou property):
Ui (X) =
inf
Z∈L1 (Ω,F ,Pi )
{Vi (Z) + E[ZX]},
X ∈ L∞ (Ω, F, Pi ),
(2.1)
where Vi is the dual Ui∗ of Ui (see (B.1)). Note how this representation of Ui can be seen as a
worst-case evaluation, based on the set of test measures Qi = {Q σ-additive measure : Q dQ
dQ
Pi , Vi ( dP
) < ∞}, where to each test measure Q ∈ Qi is associated a penalty Vi ( dP
) which
i
i
expresses the confidence of agent i on Q (the higher the penalty, the lower the confidence on
that measure).
Here we fix the space L∞ (Ω, F, Pi ) of Pi -essentially-bounded random variables as the set
of possible financial positions considered by agent i. However, we can also think of the choice
functions as defined on Lpi (Ω, F, Pi ), for any pi in [1, ∞] and possibly p1 6= p2 . Note that
we do not require the Pi ’s to fulfill any absolute continuity- or even equivalence-relation. A
priori the world views P1 and P2 are unrelated.
2.2
Formulation of the Problem
The problem we address in this paper is the optimal sharing and allocation of risk between
two agents who have different views of the world, in the sense described in Section 2.1. As
motivated in the Introduction, the discrete setting turns out to be a proper framework to
formulate this problem. Roughly speaking, no matter how different is the world view of the
two agents, we assume they agree on a finite set of possible scenarios. Therefore, any information they have about the preferences of the other, and any risk they consider or exchange,
is relative to this set. To put this into mathematical terms, we let A = {A1 , . . . , An } ⊂ F be
a finite partition of Ω and FA := σ({A1 , . . . , An }) the σ−algebra it generates. The Aj ’s are
the base events on which agents agree to exchange risk, and we assume that
Pi (Aj ) > 0 for all j = 1, . . . , n and i = 1, 2.
(2.2)
This latter condition does not only seem natural, but it is in fact necessary for the existence of
optimal allocations. Indeed, assume 0 = P1 (Aj ) < P2 (Aj ) (or vice versa, mutatis mutandis)
for some j ∈ {1, . . . , n}, then agent 2 could increase her wealth on Aj as much as she likes,
and agent 1 would take all the risk on Aj . Hence, in this situation there cannot be any
optimum. Moreover, let Q+ be the set of positive rationals, we assume that
P1 (Aj ) ∈ Q+ for all j = 1, . . . , n,
(2.3)
which is no restriction in the interesting cases. A finite partition A = {Aj }nj=1 of Ω such that
{Aj }nj=1 ⊂ F and (2.2), (2.3) hold will be called admissible. Now let A = {Aj }nj=1 be an
4
admissible partition of Ω. Then the space of admissible financial positions which the agents
consider in the exchange of risk, is the collection of all FA −measurable random variables,
that we denote by SA . The optimal risk allocation problem, for any aggregate risk X ∈ SA ,
is therefore formulated as follows:
UA (X) :=
sup
{U1 (X1 ) + U2 (X2 )}.
X1 , X2 ∈ SA ,
(2.4)
X1 + X2 = X
The solutions (X1 , X2 ) to (2.4), i.e. X1 , X2 ∈ SA such that X1 + X2 = X and UA (X) =
U1 (X1 ) + U2 (X2 ), if exist, are called optimal allocations of X. Note that, due to cashinvariance, an allocation (X1 , X2 ) ∈ SA × SA of X solves problem (2.4) if and only if it is
optimal in the sense of Pareto: for all allocations (Y1 , Y2 ) ∈ SA × SA of X s.t. Ui (Yi ) ≥
Ui (Xi ), i = 1, 2, then Ui (Yi ) = Ui (Xi ), i = 1, 2.
Clearly the space SA is isomorphic to Rn : any X ∈ SA admits a representation of the form
Pn
X = j=1 xj 1Aj , with {xj }nj=1 ⊂ R, and is identified with the vector x = (x1 , . . . , xn ) ∈ Rn .
In this way, the restriction of the agents’ choice functions Ui on SA can be read as defined
on Rn . In order to avoid confusion, we will use lowercase letters to denote functions on
Pn
Rn , i.e. ui (x1 , . . . , xn ) ≡ Ui ( j=1 xj 1Aj ). Note that the functions ui are concave, cashinvariant, normalized, and continuous (since concave and finitely-valued on Rn , see e.g. [12]
Corollary 2.3). Moreover, ui is monotone whenever Ui is monotone. We may now rewrite the
optimization problem (2.4) as follows:
u1 2u2 (x) =
sup
{u1 (x1 ) + u2 (x2 )},
n
x1 , x2 ∈ R ,
(2.5)
x1 + x2 = x
where u1 2u2 is the so-called convolution of u1 and u2 (see (B.5) and e.g. [26] for details on the
convolution operation). The function u1 2u2 : Rn → (−∞, ∞] inherits from ui the concavity
and cash-invariance properties. Moreover, due to concavity and the fact that dom(u1 2u2 ) =
dom(u1 ) + dom(u2 ), we either have u1 2u2 ≡ +∞ or u1 2u2 is finitely-valued and continuous
on Rn . If there exist optimal allocations of x ∈ Rn , i.e. problem (2.5) admits solutions,
then the convolution u1 2u2 is said to be exact at x. From now on, the penalty functions
or dual conjugates (see (B.1)) of u1 , u2 , u1 2u2 , also defined in Rn , are denoted by v1 , v2 , v
respectively. By Pi -law-invariance and Proposition A.1 we obtain the following relation:


n
X
z
j
1Aj  , ∀z ∈ Rn , i = 1, 2,
(2.6)
vi (z) = Vi 
P
(A
)
i
j
j=1
which will turn out to be useful in the proof of the main theorem.
5
2.3
Main Result
We work under the following assumptions.
Assumption 2.1. Agent 2 gives a finite penalty to the reference probability measure of agent
1, i.e.
P1 ∈ dom(v2 ),
(2.7)
where P1 is identified with the vector (p1 , . . . , pn ), with pj = P1 (Aj ) for all j = 1, . . . , n.
If we consider the dual representation of ui on Rn via its conjugate vi , which is the
analogue of representation (2.1) of Ui , Assumption 2.1 says that the reference probability P1
of agent 1 belongs to the test measures set considered by agent 2.
Note that, by Proposition A.1 and the normalization property, we always have P1 ∈
dom(v1 ), with v1 (P1 ) = 0. Therefore (2.7) implies P1 ∈ dom(v1 ) ∩ dom(v2 ), which ensures
that the convolution function u1 2u2 is finitely-valued and continuous.
Assumption 2.2. Either of the following two conditions holds:
(i) No Risk-Arbitrage (NRA), i.e. u1 2u2 (0) = u1 (0) + u2 (0) = 0,
(ii) ∂v2 (P1 ) 6= ∅.
The requirement of (NRA) can be seen as a kind of no-arbitrage condition concerning
risk, and it is exactly the equilibrium condition given in Burgert and Rüschendorf [7] (see
also [6] and [22]). It says that, in a condition of balance between demand and supply, it is
not possible to increase the utility of one agent without decreasing that of the other. This
is equivalent to the existence of a no trade equilibrium premium, the notion of which is
introduced in Deprez and Gerber [9], where Pareto optimal allocations are characterized for
convex premium principles.
The alternative condition in Assumption 2.2 is more of technical nature. Its meaning is
understood when going through the proof of Theorem 3.6. Examples 4.3,4.4 show that we
cannot expect optimal allocations in case Assumption 2.2 is not satisfied.
The main result of the paper is the following theorem, which states the existence of
optimal allocations for any risk x ∈ Rn . Its proof is prepared in Sections 3.1 and 3.2 and
finally presented in Section 3.3.
Theorem 2.3. Let A = {Aj }nj=1 be an admissible partition of Ω. Then, under Assumptions 2.1, 2.2, the convolution u1 2u2 in (2.5) is exact at any x ∈ Rn , i.e. problem (2.4)
admits solutions for every X ∈ SA .
In Section 4, we compute optimal risk allocations for prominent classes of choice functions, like the Entropic Utility (Example 4.1) and the Mean Variance Choice Function (Example 4.2). In Examples 4.1 and 4.2 optimal allocations exist for every admissible partition
6
of Ω. However, Examples 4.3 and 4.4 show that this is not the case in general, when Assumptions 2.1, 2.2 do not hold. In particular, with Example 4.4 we illustrate that the less
information is exchanged, i.e. the less base scenarios we fix, the more likely are we to find
optimal allocations. Clearly this is what we would expect.
3
Existence of Optimal Allocations
3.1
Preliminary Results on Convolution
In this section we provide results which form the basis for the proof of Theorem 2.3. Note
that Lemmas 3.2, 3.3, and 3.4 hold true in more general settings, i.e., for more general classes
of concave functions and far larger model spaces than Rn . Here, for uniformity of notation,
we enounce them in the present context.
Definition 3.1. For any non empty convex set C ⊆ Rn , the recession cone 0+ C is given by
0+ C = {y ∈ Rn : x + ty ∈ C, ∀x ∈ C, ∀t ≥ 0}.
From now on, A1 , A2 , and Au1 2u2 denote the acceptance sets of u1 , u2 , and u1 2u2 in
R respectively, i.e. A1 = {x ∈ Rn | u1 (x) ≥ 0} and similarly for u2 and u1 2u2 .
n
Lemma 3.2. For any x ∈ 0+ A1 ∩ −0+ A2 and y ∈ dom(v), we have hy, xi = 0.
Proof. If dom(v) = ∅, then there is nothing to prove. Otherwise, let x be in 0+ A1 ∩ −0+ A2
and t ≥ 0. Then tx ∈ A1 and −tx ∈ A2 , so that sx ∈ A1 + A2 ⊆ Au1 2u2 for any s ∈ R. This
implies that for all s ∈ R, 0 ≤ u1 2u2 (sx) = inf y∈Rn {v(y) + hy, sxi}, by (B.2). Hence, for any
y ∈ dom(v), shy, xi = hy, sxi ≥ −v(y) ∈ R, ∀s ∈ R, which gives hy, xi = 0, as claimed.
In the lemmas that follow, we give necessary and sufficient conditions for the exactness
of the convolution u1 2u2 .
Lemma 3.3. The convolution u1 u2 is exact at every x ∈ Rn if and only if A1 + A2 =
Au1 2u2 .
Proof. By definition of convolution, we always have A1 + A2 ⊆ Au1 2u2 . Suppose that u1 2u2
is exact and take x ∈ Au1 2u2 . Then 0 ≤ u1 2u2 (x) = u1 (y) + u2 (x − y) for some y ∈ Rn . By
cash-invariance we may assume that u1 (y), u2 (x − y) ≥ 0, thus x = y + (x − y) ∈ A1 + A2 .
Conversely, let A1 + A2 = Au1 2u2 and fix some x ∈ Rn . We have x − u1 2u2 (x) ∈ Au1 2u2
and, by hypothesis, there exists y ∈ Rn s.t. y ∈ A1 and (x − u1 2u2 (x) − y) ∈ A2 . This gives
u1 (y) + u2 (x − y) ≥ u1 2u2 (x), where the inequality is indeed an equality by definition of
convolution, and therefore the allocation (y, x − y) is optimal for x.
7
Lemma 3.4. The convolution u1 u2 is exact at every x ∈ Rn if and only if A1 + A2 is
closed.
Proof. One implication is immediate from the previous theorem. Indeed, for u1 2u2 exact,
the set (A1 + A2 ) coincides with Au1 2u2 , which is closed by continuity of u1 2u2 . For the
reverse implication, we assume that A1 +A2 is closed and claim that A1 +A2 = Au1 2u2 . Since
the inclusion A1 + A2 ⊆ Au1 2u2 is obvious, we only have to prove the reverse inclusion. To
this end, fix x ∈ Au1 2u2 and consider some maximizing sequence for the convolution in (2.5):
(yn , x − yn )n ⊂ Rn × Rn such that u1 2u2 (x) = lim {u1 (yn ) + u2 (x − yn )}. If u1 2u2 (x) > 0,
n→∞
we can find an element (yn̄ , x − yn̄ ) of the sequence such that u1 (yn̄ ) + u2 (x − yn̄ ) > 0. This
gives x = (yn̄ − u1 (yn̄ )) + (x − yn̄ + u1 (yn̄ )) ∈ A1 + A2 . On the other hand, if u1 2u2 (x) = 0,
consider the sequence (ηn1 , ηn2 )n ⊂ A1 ×A2 with ηn1 := yn −u1 (yn ) and ηn2 := x−yn −u2 (x−yn ).
We have ηn1 + ηn2 = x − u1 (yn ) − u2 (x − yn ) −→ x, which implies x ∈ A1 + A2 because
n→+∞
A1 + A2 is closed by hypothesis. This proves the equality Au1 2u2 = A1 + A2 . The exactness
then follows from Lemma 3.3.
We close this section by stating a condition that ensures the closedness of the sum of two
convex sets. The usefulness of this result is obvious by Lemma 3.4.
Lemma 3.5 (Corollary 9.1.2 in [26]). Let C1 , C2 be non-empty closed convex sets in Rn . If
there is no x 6= 0 such that x ∈ 0+ C1 and −x ∈ 0+ C2 , then C1 + C2 is closed.
3.2
Balanced Case
As preparatory result for the proof of Theorem 2.3, in this section we prove the exactness
of u1 2u2 in (2.5) in case the partition A = {A1 , . . . , An } of Ω is balanced w.r.to P1 , i.e.
P1 (Aj ) = n1 , ∀j = 1, . . . , n.
Theorem 3.6. Let A = {Aj }nj=1 be a P1 -balanced admissible partition of Ω. Then, under
Assumptions 2.1, 2.2, the convolution u1 2u2 in (2.5) is exact at any x ∈ Rn , i.e. problem
(2.4) admits solutions for every X ∈ SA .
Before we prove Theorem 3.6, we collect some helpful results. Let us first of all translate
the concept of law-invariance into this balanced discrete setting. Let Sn be the set of all
permutations in {1, . . . , n}. Since, by assumption, partition A is balanced w.r.to P1 , for any
Pn
Pn
x = (x1 , . . . , xn ) ∈ Rn and π ∈ Sn , we have that j=1 xπ(j) 1Aj and j=1 xj 1Aj have the
same law under P1 . Therefore, the P1 -law-invariance of U1 ensures that the induced function
u1 is permutation invariant on Rn : u1 (xπ ) = u1 (x) for all π ∈ Sn , where xπ is shorthand for
(xπ(1) , . . . , xπ(n) ).
The following lemma will be used in the proof of Theorem 3.6. The assertion is evident
and may be proved by induction.
8
Lemma 3.7. If for x ∈ Rn , n ≥ 2, there are i, j ∈ {1, . . . , n} such that xi > 0 > xj , then
there exist permutations π1 , . . . , πn−1 ∈ Sn such that xπ1 , . . . , xπn−1 are linearly independent.
In what follows we denote by πi,j the transposition interchanging i and j. Moreover, E
Pn
is the permutation invariant function E : Rn → R that operates ashE[x] := 1/ni· i=1 xi ,
Pn
and E is its null set E := {x ∈ Rn : E[x] = 0}. Note that E[x] = EP1
j=1 xj 1Aj .
Proof of Theorem 3.6. If n = 1, exactness of u1 2u2 follows from cash-invariance. Henceforth,
let n ≥ 2. If there is no x ∈ Rn \{0} such that x ∈ 0+ A1 ∩ −0+ A2 , then the exactness
follows from Lemma 3.5 and Lemma 3.4. Now suppose there exists x 6= 0 in 0+ A1 ∩ −0+ A2 .
From Assumption 2.1 and Lemma 3.2, we have that E[x] = 0. Consequently, there are
i, j ∈ {1, . . . , n} such that xi > 0 > xj . Moreover, x ∈ E ∩ 0+ A1 . We claim that this implies
the existence of a vector x0 ∈ E ∩ 0+ A1 , x0 6= 0, having the following property:
for all π ∈ Sn there exists µ ∈ Sn such that −x0π = x0µ .
(3.1)
Suppose that such an x0 exits. Then, since E[x0 ] = 0, there are i, j ∈ {1, . . . , n} such that
x0i > 0 > x0j . Hence, by Lemma 3.7, there exist n − 1 linearly independent permutations
x0π1 , . . . , x0πn−1 of x0 . Therefore, x0π1 , . . . , x0πn−1 ∈ E ∩ 0+ A1 form a basis of the (n − 1)n−1
dimensional subspace E. Now, choose any y ∈ E. For appropriate {ai }i=1
⊂ R and {µi }n−1
i=1 ⊂
Sn , we have
y=
n−1
X
ai x0πi =
i=1
n−1
X
|ai |x0µi ∈ E ∩ 0+ A1 ,
i=1
+
due to (3.1) and the fact that E ∩ 0 A1 is a permutation invariant convex cone. Thus
E = E ∩ 0+ A1 . From Proposition A.1, E[x] ≥ u1 (x) ∀x ∈ Rn and then A1 ⊆ AE . On the
other hand, we always have 0+ A1 ⊆ A1 , so that 0+ A1 ⊆ A1 ⊆ AE . Now we are going to
prove that these are all equalities, showing that AE ⊆ 0+ A1 . Indeed, fix x ∈ AE and consider
any y ∈ A1 and t ≥ 0. Then u1 (y +tx) = u1 (y +tE[x]+t(x−E[x])) ≥ 0 follows from E[x] ≥ 0
and (x − E[x]) ∈ E ⊆ 0+ A1 . Therefore x ∈ 0+ A1 and AE ⊆ 0+ A1 , which implies u1 = E.
Now, by (B.7) we have u1 2u2 = E + v2 (P1 ). Thus, if condition (i) of Assumption 2.2 holds,
then v2 (P1 ) = 0 and u1 2u2 = E = u1 , which in particular ensures the exactness of the
convolution. On the other hand, if condition (ii) of Assumption 2.2 is satisfied, then for any
x ∈ Rn and y ∈ −∂v2 (P1 ) we obtain u1 2u2 (x) = E[x−y]+E[y]+v2 (P1 ) = u1 (x−y)+u2 (y),
by (B.4). Therefore, the convolution is exact in this case too.
Finally, in order to verify that there indeed exists an 0 6= x0 ∈ E ∩ 0+ A1 satisfying (3.1),
let 0 6= x ∈ E ∩ 0+ A1 and note that (3.1) is always true for n ≤ 2. In case n > 2, on the
other hand, we consider the following algorithm:
9
input: 0 6= x ∈ E ∩ 0+ A1
for i = 3 to n:
if ∀ π ∈ Sn ∃ µ ∈ Sn : −xπ = xµ then return x0 := x end,
else sort x in such a way that the output x̄ satisfies x̄i−2 > 0 and x̄i−1 < 0, and
additionally x̄j = 0 for all j < i − 2 in case i > 3.
x̂ := x̄ +
−x̄i−2
x̄πi−2,i−1
x̄i−1
x := x̂ .
end
Since E ∩0+ A1 is a permutation invariant convex cone, sorting 0 6= x ∈ E ∩0+ A1 as described
in the algorithm, the output x̄ of each cycle is still an element of E ∩ 0+ A1 , as is x̄πi−2,i−1 .
i−2
+
+
From −x
xi−1 > 0, then also x̂ ∈ E ∩ 0 A1 , thus the algorithm never leaves the set E ∩ 0 A1 .
Furthermore, in each cycle the algorithm either terminates or eliminates the i − 2-nd entry,
that is, it builds a vector x satisfying xi−2 = 0. Since for j = i, . . . , n:
x̄j 6= 0
and
x̄i−1 −
⇒
x̄2i−2
=0
x̄i−1
x̄j −
⇔
x̄i−2
x̄j 6= 0
x̄i−1
x̄i−2 = −x̄i−1 ,
our algorithm does not return the zero vector at any cycle. Indeed, suppose for the moment it did return the zero vector. Then the preceding relations tell us that x̄j = 0 for
all j ∈ {1, . . . , n} \ {i − 2, i − 1} and x̄i−2 = −x̄i−1 = a for some a 6= 0. Thus x̄ is of
type (x̄1 , . . . , x̄i−3 , x̄i−2 , x̄i−1 , x̄i , . . . , x̄n ) = (0, . . . , 0, a, −a, 0, . . . , 0), which cannot happen
because this implies that the outcome of the previous cycle did satisfy the breaking condition (3.1), and thus the algorithm should already have terminated. Moreover, in case the
algorithm does not terminate before all possible n − 2 cycles are through, it returns a vector
of type (0, . . . , 0, a, −a) for some a ∈ R\{0}. So finally we find an 0 6= x0 ∈ E ∩ 0+ A1 which
does satisfy (3.1).
3.3
Proof of Theorem 2.3
We reduce the general discrete case to a balanced setting in order to apply Theorem 3.6.
Remember that, by admissibility, the probabilities ai := P1 (Ai ) are in Q+ for all i = 1, . . . , n,
and consider the greatest rational number a s.t. ai /a are all integers for i = 1, . . . , n. By
the non-atomicity of (Ω, F, P1 ) and (Ω, F, P2 ), for each i = 1, . . . , n we can find a partition
{Bi1 , . . . , Bimi } ⊂ F of the event Ai such that
P1 (Bij ) =
P2 (Ai )
P1 (Ai )
= a and P2 (Bij ) =
,
mi
mi
10
(3.2)
where mi := ai /a (see e.g. [10] Corollary 1.1). Therefore, we end up with a P1 -balanced
admissible partition B = {Bij , j = 1, . . . , mi , i = 1, . . . , n} of Ω, which is a refinement of
partition A and is composed of M := 1/a sets. Denoting by FB the σ-algebra generated
by partition B, and SB the space of FB -measurable random variables, we clearly have the
inclusions FA ⊆ FB ⊆ F and SA ⊆ SB . Note that P1 and P2 restricted to FB are equivalent
(the same as in FA ). Moreover, we have that the densities on FA and FB are respectively
fA :=
and
n
X
P2 (Ai )
dP2 1A
=
dP1 FA
P (Ai ) i
i=1 1
mi
mi
n X
n X
X
X
P2 (Ai )
dP2 P2 (Bij )
1Bij =
1B = fA ,
fB :=
=
dP1 FB
P
(B
)
P (Ai ) ij
ij
i=1 j=1 1
i=1 j=1 1
by (3.2). Therefore, for any FB -measurable r.v. ξ we have
fB EP2 [ξ|FA ] = EP1
ξ FA = EP1 [ξ|FA ].
fA
(3.3)
Now fix X ∈ SA and consider the maximization problem restricted to the FB -measurable
pairs:
UB (X) :=
sup
{U1 (X1 ) + U2 (X2 )}.
X1 , X2 ∈ SB ,
(3.4)
X1 + X2 = X
Suppose Assumptions 2.1 and 2.2 are satisfied w.r.to partition B. Then, from Theorem 3.6 we
know that problem (3.4) admits solutions for any FB -measurable total risk, thus in particular
for the FA -measurable r.v. X we have fixed. Let (X1 , X2 ) ∈ SB × SB be such a solution. On
the one hand, from equality (3.3) we have
EP1 [X1 |FA ] + EP2 [X2 |FA ] = EP1 [X1 + X2 |FA ] = EP1 [X|FA ] = X.
On the other hand, by Proposition A.1, the Pi -law-invariance of Ui implies Ui (EPi [Xi |FA ]) ≥
Ui (Xi ), so that (EP1 [X1 |FA ], EP2 [X2 |FA ]) is an FA × FA -measurable allocation of X which
is at least as good as (X1 , X2 ). Thus we may assume that (X1 , X2 ) ∈ SA × SA . Hence,
U1 (X1 ) + U2 (X2 ) = UB (X) ≥ UA (X) ≥ U1 (X1 ) + U2 (X2 ),
so we obtain UA (X) = U1 (X1 ) + U2 (X2 ), i.e. the exactness of u1 2u2 in (2.5). It remains to
Pn Pmk
prove that Assumptions 2.1 and 2.2 are indeed satisfied by uB
i (x) ≡ Ui (
k=1
j=1 xkj 1Bkj ),
∗
x ∈ RM , and viB = (uB
i ) , i = 1, 2. To this end, note that, according to (2.6), by (3.2) we
have

v2B (P1 )
=
v2B (a, . . . , a)
= V2 
mk
n X
X
k=1 j=1

a
1B  = V2
P2 (Bkj ) kj
11
n
X
mk a
1A
P2 (Ak ) k
k=1
!
= v2 (P1 ),
so P1 ∈ dom(v2B ) whenever P1 ∈ dom(v2 ). Moreover, it is easily verified that y ∈ ∂v2 (P1 )
Pn Pmk
yk 1Bkj in RM is an element of ∂v2B (P1 ).
implies that the vector corresponding to k=1 j=1
B
Finally, u1 2u2 (0) = 0 implies uB
1 2u2 (0) = 0, because otherwise there would be Y ∈ SB
such that U1 (Y ) + U2 (−Y ) > 0. But then again by Proposition A.1 and (3.3) we may assume
that (Y, −Y ) ∈ SA × SA which implies u1 2u2 (0) > 0 and this is a contradiction. Therefore,
Assumptions 2.1 and 2.2 hold for the admissible partition B of Ω as well, and this concludes
the proof.
4
Examples
In what follows we provide some examples of optimal risk sharing problems, including prominent choice functions such as the Entropic Utility and the Mean Variance Choice Function.
One of the main ingredients we will use to study the existence of optimal allocations and, in
case, to compute them explicitly, is Proposition A.2, which plays an important role whenever
we have some information about the gradient of the convolution function.
In Examples 4.1, 4.2 the hypothesis of Theorem 2.3 are satisfied for every admissible
partition of Ω, so there always exist Pareto optimal allocations, and we are able to compute
them. Actually we can say even more: in these examples, we can formulate and solve the
optimal risk sharing problem in continuous setting too, provided some strong link between
the different world views.
Example 4.1. (Convolution of Entropic Utilities) The Entropic Utility w.r.to Pi , with parameter β > 0 (see e.g. [20] and [21]) is given by
n
o
h Xi
Entrβi (X) = −β log EPi e− β = inf EQ [X] + βH(Q|Pi ) : Q Pi ,
X ∈ L1 (Ω, F, Pi ),
h i
dQ
where, for any Q Pi , H(Q|Pi ) = EQ log dP
denotes the relative entropy of Q w.r.to
i
i
i
Pi . The dual conjugate of Entrβ is given by Vβ = βH(.|Pi ) on the set of probability measures
Q s.t. Q Pi , and Vβi = +∞ otherwise (see e.g. [18]). Let β1 , β2 > 0 and Ui = Entrβi i ,
i = 1, 2. Note that, for any admissible partition A of Ω, Assumptions 2.1 and 2.2 (ii) are
satisfied. Hence, for every X ∈ SA there is an optimal allocation (X1 , X2 ) ∈ SA × SA and
it is given in (4.2) below with f =
dP1
dP2 |FA .
However, for this kind of choice functions we
can show even more. Suppose for the moment that P1 ≈ P2 with density
p
p
dP1
dP2
bounded and
p
bounded away from 0. Then, ∀p ∈ [1, ∞], L := L (Ω, F, P1 ) = L (Ω, F, P2 ) and the risk
sharing problem is well defined on Lp as well, and formulated as
U1 2U2 (X) =
sup
U1 (X1 ) + U2 (X2 ),
p
X1 , X 2 ∈ L ,
X1 + X2 = X
12
X ∈ Lp .
(4.1)
In the following we will show that (4.1) too admits solutions for every X ∈ Lp . Note that
here we require a strong relation between the world views P1 and P2 which we do not in
the discrete setting. Denote by f the density
dP1
dP2
and consider the bidual (U1 2U2 )∗∗ of the
convolution in (4.1) (see (B.2) and (B.6)):
(U1 2U2 )∗∗ (X)
=
inf {EQ [X] + β1 H(Q|P1 ) + β2 H(Q|P2 )}
dQ
dQ
+ β2 log
inf EQ [X] + EQ β1 log
QPi
dP1
dP2
dQ
β1 X + β1 β2 log(dP1 /dP2 )
β1 + β2
inf EQ
+ β1 EQ log
β1 QPi
β1 + β2
dP1
β1 X + β1 β2 log(f )
β1 + β2
U1
β1
β1 + β2
β1 + β2
β2 X − β1 β2 log(f )
U2
.
β2
β1 + β2
QPi
=
=
=
=
Now, for any aggregate risk X, by choosing
X1 :=
β1 X + β1 β2 log(f )
,
β1 + β2
X2 := X − X1 =
β2 X − β1 β2 log(f )
,
β1 + β2
(4.2)
we obtain (U1 2U2 )∗∗ (X) = U1 (X1 ) + U2 (X2 ), which in view of (B.3) implies U1 2U2 (X) =
U1 (X1 )+U2 (X2 ). Therefore, the risk sharing problem (4.1) admits solutions for every X ∈ Lp ,
for any p ∈ [1, ∞].
Clearly the very same computation can be done for the bidual of the convolution in the
discrete setting (2.5), for any admissible partition A of Ω. Therefore, for any X ∈ SA , we
have that an optimal allocation is given by (4.2), keeping in mind that in this case f means
dP1
3
dP2 |FA .
Example 4.2. (Convolution of Mean Variance Choice Functions) The Mean Variance Choice
Function w.r.to Pi , with parameter γ > 0, is given by
M Vγi (X) = EPi [X] − γEPi [(X − EPi [X])2 ],
and its dual by Vγi (Z) =
X ∈ L1 (Ω, F, Pi ),
EPi [(Z−1)2 ]
,
4γ
for all Z ∈ L2 (Ω, F, Pi ) with EPi [Z] = 1, and Vγi (Z) =
+∞ otherwise (see e.g. [1]). Let γ1 , γ2 > 0 and Ui = M Vγii , i = 1, 2. As in the previous
example, Assumptions 2.1 and 2.2 (ii) are satisfied for every admissible partition A of Ω, so
the existence of optimal allocations follows by Theorem 2.3. Again, we can show even more.
Indeed, if we consider the continuous setting described in Example 4.1, we can prove the
exactness of the risk sharing problem in Lp as well, for all p ∈ [1, ∞]. In particular, by means
of Proposition A.2, we have that an optimal allocation of any given total risk X ∈ SA or
X ∈ Lp (i.e. when the problem is considered in discrete or in continuous setting) is given by
X1 :=
γ2
X,
γ2 + γ1 f
X2 :=
13
γ1 f
X,
γ2 + γ1 f
recalling the different meaning of f in the two cases.
3
The following example shows that, in general, we cannot expect existence of optimal
allocations if Assumption 2.2 is not satisfied. Moreover, we present a continuous setting
which does not allow for optimal allocations, whereas there are admissible partitions such
that problem (2.4) admits solutions.
dP1
1
3
1
Example 4.3. Let P1 ≈ P2 be such that P2 dP
=
=
P
=
= 12 . Then, obvi2
dP2
2
dP2
2
ously, Lp (Ω, F, P1 ) = Lp (Ω, F, P2 ) =: Lp for all p ∈ [1, ∞]. Now fix p ∈ [1, ∞] and let
U1 (X) = EP1 (X), X ∈ Lp
and U2 (X) =
1
EP2 [X] + Entr12 (X) , X ∈ Lp .
2
The dual conjugate of U1 is V1 = δ(·|{P1 }), which equals zero on P1 and +∞ elsewhere. The
dual of U2 is given by
1
1
1
− ZQ X − log EP2 e−X
= EP2 [(2ZQ − 1) log(2ZQ − 1)]
V2 (Q) = sup EP2
2
2
2
X∈Lp
if ZQ := dQ/dP2 ≥ 1/2, and V2 (Q) = +∞ elsewhere. Moreover, we have that ∂V2 (Q) =
{log(2ZQ − 1) + c : c ∈ R} when well defined, and ∅ elsewhere. Hence, P1 ∈ dom(V2 ),
but ∂V2 (P1 ) = ∅. Here the convolution in Lp leads to U1 2U2 (X) = EP1 (X) + V2 (P1 ), with
∂U1 2U2 (X) = {P1 } ∀X ∈ Lp . Therefore, by Proposition A.2 there are no solutions to the
risk sharing problem in Lp .
Now fix any admissible partition A = {A1 , . . . , An } of Ω. By (2.6) we have that
)
(
n
X
P2 (Ai )
n
∀i = 1, . . . , n
dom(v2 ) = z ∈ R :
zi = 1, zi ≥
2
i=1
and, for z ∈ dom(v2 ),
n
1 X
2zi
v2 (z) =
2zi − P2 (Ai ) log
−1 .
2 i=1
P2 (Ai )
Thus ∂v2 (z) 6= ∅ if and only if 2zi > P2 (Ai ) for all i = 1, . . . , n, and, provided this holds,
2zn
2z1
− 1 + c, . . . , log
−1 +c :c∈R .
(4.3)
∂v2 (z) =
log
P2 (A1 )
P2 (An )
Since
dP1
dP2 |FA
≥ 12 , we always have P1 ∈ dom(v2 ). Moreover, by Proposition A.2 in conjunction
with the fact that ∂u1 2u2 (x) = {P1 } for all x ∈ Rn , we obtain that u1 2u2 is exact if and
only ifnconditiono(ii) of Assumption 2.2 holds. Hence, if there is one j ∈ {1, . . . , n} such that
dP1
= 12 a.s., then ∂v2 (P1 ) = ∅, so there are no optimal allocations. Otherwise, if
Aj ⊆ dP
2
n
o
dP1
1
there is no j ∈ {1, . . . , n} such that Aj ⊆ dP
=
a.s., then Assumptions 2.1 and 2.2
2
2
are satisfied, and according to (4.3) the optimal risk allocations of any X ∈ SA are given by
14
(X − Y, Y ) with Y = − log(2 ·
dP1
dP2 |FA
− 1) + c, c ∈ R. Note how the share of agent 2 does
not depend on the total risk X. Indeed, what she takes is some measure of the difference
1
between the world views. Clearly, this is equal to zero when dP
3
dP2 |FA ≡ 1.
In Example 4.4 we motivate that the smaller the set of base scenarios, the more likely are
we to find optimal solutions to the risk sharing problem. Obviously, if there is only one base
scenario, i.e. if only cash is exchanged, then, due to cash-invariance, any allocation (x − y, y),
y ∈ R, is an optimal allocation of x ∈ R.
Example 4.4. Let A = {A1 , A2 , A3 } be a P1 -balanced partition of Ω and
(
)
3
3
X
X
dQ
1
1 2
C := Q P1 :
=
3ai 1Ai ,
.
(ai − ) ≤
dP1
3
6
i=1
i=1
Let Π : C → R3 be given by Π(Q) = (Q(A1 ), Q(A2 ), Q(A3 )) and consider the following
function
1
α(y) = √ −
6
3
1 X
1
−
(yi − )2
6 i=1
3
! 12
(
,
y ∈ B :=
3
X
1
1
z ∈ R3 :
(zi − )2 ≤
3
6
i=1
)
.
Note that α is not (sub)differentiable on the boundary of B. Let U1 be given by
U1 (X) := inf EQ [X] + V1 (Q),
QP1
X ∈ L∞ (Ω, F, P1 ),
where V1 (Q) = α(Π(Q)) + δ(Q|C). Moreover, let P2 be the probability measure given by
dP2
dP1
= 21A1 + 12 1A2 + 12 1A3 and let U2 (X) := EP2 (X), X ∈ L∞ (Ω, F, P2 ). Suppose that agents
1 and 2, endowed with choice functions U1 and U2 , agree on the base scenarios {A1 , A2 , A3 }.
Then we have
A
uA
1 (x) = inf3 hz, xi + v1 (z)
and
z∈R
uA
2 (x) =
1
1
2
x1 + x2 + x3 ,
3
6
6
x ∈ R3 ,
where v1A = α + δ(·|Π(C)). Note that P2 = (2/3, 1/6, 1/6) ∈ dom(v1A ). Thus
A
A
A
uA
1 2u2 (x) = u2 (x) + v1 (P2 )
and
A
3
∂uA
1 2u2 (x) = {P2 }, ∀ x ∈ R .
Since (2/3, 1/6, 1/6) is a boundary point of B, we have that ∂v1A (P2 ) = ∅ and then, by
Proposition A.2, there are no optimal allocations (note that (NRA) is not satisfied because
√
e = {A1 ∪
v1A (P2 ) = 1/ 6). However, if we suppose that the agents agree on the scenarios A
A2 , A3 }, then we obtain that
A
uA
1 (x) = sup hz, xi + v1 (z),
e
e
z∈R2
x ∈ R2 ,
5
1
A
where v1A (z) = v1A ( z21 , z21 , z2 ), and uA
2 (x) = 6 x1 + 6 x2 . Again P2 = (5/6, 1/6) ∈ dom(v1 ),
e
5 1
5
, 12
, 6 ). Hence there are
but this time ∂v2A (5/6, 1/6) 6= ∅ because α is differentiable at ( 12
optimal allocations for any total risk X ∈ SAe.
3
e
e
e
15
Remark 4.5. Note that in Examples 4.1,4.2 the optimal allocations are obtained as functions
of the total risk and of the density function. Indeed, with the same arguments as in the proof
of Theorem 2.3, for any total risk X and any allocation (X1 , X2 ) of X, we have that the
allocation
(EP1 [X1 |σ(X, f )], EP2 [X2 |σ(X, f )])
of X outperforms (X1 , X2 ), thus the optimization problem can be formulated over the
3
σ(X, f )-measurable allocations.
As we have seen, in some circumstances it is possible to treat some continuous cases
together with the discrete ones and use Proposition A.2 to possibly characterize optimal
allocations. Clearly this is not the general situation (see e.g. Examples 4.3,4.4). However,
there are cases where, even if we cannot state the existence of solutions or we are not able
to compute them, our results can be used for approximations. Indeed, assume the density
dP1 /dP2 to be simple, p ∈ [1, ∞) and the choice functions Ui : Lp → [−∞, ∞) to satisfy
(C1)–(C5) on Lp . Assumptions 2.1 and 2.2 here become P1 ∈ dom(V2 ), and U1 U2 (0) = 0 or
∂V2 (P1 ) 6= ∅ respectively, with V2 conjugate of U2 . Let U1 be monotone and continuous, which
ensures U1 U2 to be continuous as well. Fix X ∈ Lp and consider simple r.v.’s {Xn }n∈N
converging to X in the Lp -norm. According to Remark 4.5, we obtain that the optimization
problem in Lp is exact at Xn , with some simple optimal allocation (Y1n , Y2n ), for all n ∈ N.
Therefore, by continuity of U1 and U1 U2 , we have U1 (X − Y2n ) + U2 (Y2n ) −→ U1 U2 (X),
n→+∞
i.e., the optimal value can be approximated by means of solutions to discrete problems.
What is worth noticing here is the fact that we approach the optimal value by allocations
that in many cases can be explicitly computed by means of standard algorithms (for convex
optimization in Rn ).
A
Some Useful Results
In this section we collect some known results which are used throughout the paper.
e G, P) be a non-atomic standard probability space, p ∈
Proposition A.1 ([18],[15]). Let (Ω,
p e
[1, ∞] and U : L (Ω, G, P) → [−∞, +∞) be a P−law-invariant proper concave u.s.c. function.
Then, for any sub-σ-algebra B ⊆ G,
U (EP [X|B]) ≥ U (X),
e G, P).
∀X ∈ Lp (Ω,
In particular, U (EP (X)) ≥ U (X).
e G, P) be a probability space, p ∈ [1, ∞] and U1 , U2 :
Proposition A.2 ([24]). Let (Ω,
p e
e G, P) s.t.
L (Ω, G, P) → [−∞, ∞) be proper concave u.s.c. functions. Then, for X ∈ Lp (Ω,
16
e G, P) × Lp (Ω,
e G, P) of X,
∂U1 2U2 (X) 6= ∅ and for any allocation (X1 , X2 ) ∈ Lp (Ω,
U1 2U2 (X) = U1 (X1 ) + U2 (X2 )
B
⇐⇒
∂U1 2U2 (X) = ∂U1 (X1 ) ∩ ∂U2 (X2 ).
Some Functional Analysis
In this section we recall some well-known concepts and results from convex analysis. Let H
be some locally convex space. Given a concave function ϕ : H → [−∞, +∞), its conjugate
ϕ∗ : H∗ → [ϕ(0), +∞] is defined as
ϕ∗ (µ) := sup {ϕ(X) − hµ, Xi},
(B.1)
X∈H
and it is a convex and σ(H∗ , H)-lower semi-continuous function on H∗ which is proper if and
only if ϕ is proper. The conjugate ϕ∗∗ : H → [−∞, +∞) of ϕ∗ is
ϕ∗∗ (X) := inf ∗ {ϕ∗ (µ) + hµ, Xi},
µ∈H
(B.2)
and it is a concave, σ(H, (H∗ )-u.s.c. function on H, with
ϕ∗∗ ≥ ϕ,
(B.3)
and ϕ∗∗ = ϕ if and only if ϕ is proper and σ(H, H∗ )-u.s.c. or ϕ = ±∞.
The gradient of ϕ at X ∈ H is given by
∂ϕ(X) = {µ ∈ H∗ : ϕ(Y ) ≤ ϕ(X) + hµ, Y − Xi, ∀Y ∈ H}
and the gradient of ϕ∗ at µ ∈ H∗ by
∂ϕ∗ (µ) = {X ∈ H : ϕ∗ (ν) ≥ ϕ∗ (µ) + hν − µ, Xi, ∀ν ∈ H∗ }.
For ϕ proper, concave and u.s.c., the following chain of equivalences holds for any pair
(X, µ) ∈ H × H∗ :
µ ∈ ∂ϕ(X) ⇔ X ∈ −∂ϕ∗ (µ) ⇔ ϕ(X) = ϕ∗ (µ) + hµ, Xi.
(B.4)
Let φ : H → [−∞, +∞) be another concave function. The convolution ϕ2φ of ϕ and φ is
given by
ϕ2φ(X) := sup ϕ(X − Y ) + φ(Y ).
(B.5)
Y ∈H
It inherits concavity from ϕ and φ. Moreover, its conjugate satisfies
(ϕ2φ)∗ = ϕ∗ + φ∗ .
(B.6)
In particular, if the convolution function is proper and u.s.c., then by (B.2) it admits the
following dual representation
ϕ2φ(X) = inf ∗ {ϕ∗ (µ) + φ∗ (µ) + hµ, Xi}.
µ∈H
17
(B.7)
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